Concentration in Gossip Opinion Dynamics over Random Graphs

TL;DR

High-probability bounds are derived for the distance between time-averaged opinions and the expected final opinions over the expected graph with the help of concentration inequalities for Markov chains.

Abstract

We study concentration inequalities in gossip opinion dynamics over random graphs. In the model, a network is generated from a random graph model with independent edges, and agents interact pairwise randomly over the network. During the process, regular agents average neighbors' opinions and then update, whereas stubborn agents do not change opinions. To approximate the original process, we introduce a gossip model over an expected graph, obtained by averaging all possible networks generated from the random graph model. Using concentration inequalities, we derive high-probability bounds for the distance between the expected final opinion vectors over the random graph and over the expected graph. Leveraging matrix perturbation results, we show how such concentration can help study the effect of network structure on the expected final opinions in two cases: (i) When the influence of stubborn agents is large, the expected final opinions polarize and are close to stubborn agents' opinions. (ii) When the influence of stubborn agents is small, the expected final opinions are close to each other. With the help of concentration inequalities for Markov chains, we obtain high-probability bounds for the distance between time-averaged opinions and the expected final opinions over the expected graph. In simulation, we validate the theoretical findings, and study a gossip model over a stochastic block model that has community structure.

Figures (7)

• Figure 1: \newlabelfig:exp0 Different categories of opinion distributions (terminology from devia2022framework). (a) Perfect consensus in severity of climate change, where $0$ means "don't know", $1$ "not serious", $2$ "fairly serious", and $3$ "very serious". Almost all respondents in Spain of a survey regard climate change as a very serious problem eurobarometer2020attitudes. (b) Polarization of pro-enviormental votes on legislation from U.S. senators in 2015, where Democrats show high percentage of pro-environmental votes but Republicans show low percentage. The political elites hold extreme positions in line with their parties (Source: League of Conservation Voters) dunlap2016political. (c) Clustering of opinions on whether people should maintain their distinct cultural identities downey2001attitudinal. Three clusters can be observed on the left, middle, and right, respectively. (d) Dissensus of French political opinions from European Social Survey 2012. Individual opinions are diverse, with most of them held by a non-negligible number of people.
• Figure 1: Illustration of a gossip model over an RG-S and a gossip model over an expected graph. On the top left of the figure, a random graph $\mathcal{G}$ is constructed from an RG-S. Circles and squares represent regular and stubborn agents, respectively. On the top middle, a gossip model evolves over $\mathcal{G}$, where a single existing edge is selected at each time. On the top right, the expression of the expected final opinion vector is given. On the bottom left, the expected graph $\bar{\mathcal{G}}$ is obtained by averaging the random graph $\mathcal{G}$. On the bottom middle, a gossip model evolves over the expected graph, where an edge is selected with probability proportional to its weight in the expected adjacency matrix. On the bottom right, the expression of the expected final opinion vector over the expected graph is given.
• Figure 1: Comparison of the theoretical bound provided by \ref{['thm:concentration_states']} (i) with simulation. A log-log plot is given in the figure.
• Figure 2: A sample of an SBM. In the graph, dots and squares represent regular and stubborn agents, respectively.
• Figure 3: \newlabelfig:phase_transition0 The profile of the expected final opinions $\mathbf{x}^{\mathcal{G},n}$ under different stubborn influence. The dashed lines represent the three distinct values of $\mathbf{x}^{*,n}$ corresponding to the communities.

Theorems & Definitions (37)

• Example 1.1
• Definition 2.1: Random graph model
• Example 2.2
• Proposition 2.3: Stability and limit theorems
• Definition 2.4: Random graph with stubborn agents, RG-S
• Definition 2.5: Gossip model over expected graph
• Remark 4.2
• Theorem 4.3: Concentration of expected final opinions
• Proof 1
• Remark 4.4